BayesianProject

[Obtain the dataset at this address] (https://www.kaggle.com/datasets/everydaycodings/produce-prices-dataset)

Exploratory Analysis & Data Cleanup

Let’s do some exploratory data analysis and data cleanup to start things off.

produce <- read.csv("ProductPriceIndex.csv")
print(summary(produce))

 productname            date            farmprice         atlantaretail      chicagoretail      losangelesretail  
 Length:15766       Length:15766       Length:15766       Length:15766       Length:15766       Length:15766      
 Class :character   Class :character   Class :character   Class :character   Class :character   Class :character  
 Mode  :character   Mode  :character   Mode  :character   Mode  :character   Mode  :character   Mode  :character  
 newyorkretail      averagespread     
 Length:15766       Length:15766      
 Class :character   Class :character  
 Mode  :character   Mode  :character  

print(head(produce))

Notice that all the features are characters. Some of them, like “farmprice,” would make more sense as floats; additionally, the features that represent prices have a pesky dollar sign in front. So let’s fix that first.

produce$farmprice <- as.numeric(gsub("\\$", "", produce$farmprice))
produce$atlantaretail <- as.numeric(gsub("\\$", "", produce$atlantaretail))
produce$chicagoretail <- as.numeric(gsub("\\$", "", produce$chicagoretail))
produce$losangelesretail <- as.numeric(gsub("\\$", "", produce$losangelesretail))
produce$newyorkretail <- as.numeric(gsub("\\$", "", produce$newyorkretail))
produce$averagespread <- as.numeric(gsub("\\%", "", produce$averagespread))

Warning: NAs introduced by coercion

print(head(produce))

Awesome. Another cleanup we’ll perform is adjusting the “date” feature to date-time format, which will make it easier to perform computations on.

produce$date <- as.POSIXct(produce$date, format = "%Y-%m-%d")
print(head(produce))

Perfect. Finally, let’s remove all the null values, if any.

produce <- na.omit(produce)

Now we’re done with our initial data cleanup. We model the variable “averagespread” based on one quantitative predictor: “farmprice”.

library(ggplot2)

ggplot(produce, aes(x = farmprice, y = averagespread)) +
  geom_point() +
  labs(title = "Scatterplot of Farm Price vs. Average Spread",
       x = "Farm Price",
       y = "Average Spread") +
  theme_minimal() +
  geom_point(size = 0.1)

This is a pretty strong graph, but let’s check out the other variables before building a regression model. We perform feature engineering by taking the average retail price, then we model “averagespread” based on the new feature.

produce$avgretail <- rowMeans(produce[, c("atlantaretail", 
                                          "chicagoretail", 
                                          "losangelesretail", 
                                          "newyorkretail")], 
                              na.rm = TRUE)
ggplot(produce, aes(x = farmprice, y = avgretail)) +
  geom_point() +
  labs(title = "Scatterplot of Farm Price vs. Average Retail",
       x = "Farm Price",
       y = "Average Retail Price") +
  theme_minimal() +
  geom_point(size = 0.1)

This doesn’t look very promising, so let’s make the same model but we’ll isolate specific products.

#Scatterplot for strawberries
produce_strawberries <- subset(produce, productname == "Strawberries")
ggplot(produce_strawberries, aes(x = farmprice, y = avgretail)) +
  geom_point() +
  labs(title = "Scatterplot of Farm Price vs. Average Retail Price (Strawberries)",
       x = "Farm Price",
       y = "Average Retail Price") +
  theme_minimal() +
  geom_point(size = 0.1)

#Scatterplot for potatoes
produce_potatoes <- subset(produce, productname == "Potatoes")
ggplot(produce_potatoes, aes(x = farmprice, y = avgretail)) +
  geom_point() +
  labs(title = "Scatterplot of Farm Price vs. Average Retail Price (Potatoes)",
       x = "Farm Price",
       y = "Average Retail Price") +
  theme_minimal() +
  geom_point(size = 0.1)

#Scatterplot for oranges
produce_oranges <- subset(produce, productname == "Oranges")
ggplot(produce_oranges, aes(x = farmprice, y = avgretail)) +
  geom_point() +
  labs(title = "Scatterplot of Farm Price vs. Average Retail Price (Oranges)",
       x = "Farm Price",
       y = "Average Retail Price") +
  theme_minimal() +
  geom_point(size = 0.1)

While we can see some trends, it might be more worth our while to model “farmprice vs averagespread” for each product.

#Scatterplot for strawberries
ggplot(produce_strawberries, aes(x = farmprice, y = averagespread)) +
  geom_point() +
  labs(title = "Scatterplot of Farm Price vs. Average Spread (Strawberries)",
       x = "Farm Price",
       y = "Average Retail Price") +
  theme_minimal() +
  geom_point(size = 0.1)

#Scatterplot for potatoes
ggplot(produce_potatoes, aes(x = farmprice, y = averagespread)) +
  geom_point() +
  labs(title = "Scatterplot of Farm Price vs. Average Spread (Potatoes)",
       x = "Farm Price",
       y = "Average Retail Price") +
  theme_minimal() +
  geom_point(size = 0.1)

#Scatterplot for oranges
ggplot(produce_oranges, aes(x = farmprice, y = averagespread)) +
  geom_point() +
  labs(title = "Scatterplot of Farm Price vs. Average Spread (Oranges)",
       x = "Farm Price",
       y = "Average Retail Price") +
  theme_minimal() +
  geom_point(size = 0.1)

Linear Regression Model

These plots demonstrate that “farmprice vs averagespread” has a distinct trend resembling the graph 1/x. Let’s compare 2 models: a linear regression model and a hierarchical model, starting with the former. We’ll perform one more data cleanup: we’ll remove all the rows where “farmprice” = 0 because our model will encounter an error for those values, since 1/0 is undefined.

produce <- produce[produce$farmprice != 0, ]

# install.packages("bayesrules")
library(bayesrules)
# install.packages("tidyverse")
library(tidyverse)

# install.packages("bayesplot")
library(bayesplot)

# install.packages("tidybayes")
library(tidybayes)
# install.packages("rstanarm")
library(rstanarm)

# install.packages("broom.mixed")
library(broom.mixed)
# install.packages("gridExtra")
library(gridExtra)

library(dplyr)
produce_model <- stan_glm(
  averagespread ~ I(1/farmprice) + productname,
  data = produce, family = gaussian, 
  prior_intercept = normal(25, 5),
  prior = normal(0, 2.5, autoscale = TRUE), 
  prior_aux = exponential(1, autoscale = TRUE),
  chains = 4, iter = 5000*2, seed = 84735,
  prior_PD = FALSE)

Regression parameters:

head(as.data.frame(produce_model), 3)

Posterior regression structure: scatterplot of farmprice vs average spread

produce %>%
  add_fitted_draws(produce_model, n = 100) %>%
  ggplot(aes(x=farmprice, y=averagespread, color=productname)) +
    geom_line(aes(y = .value,  alpha = .1,
                  group = paste(productname, .draw))) +
    geom_point(data = produce, size = 0.1)

The structure seems to mimic the trend in the original scatterplot.

Plotting posterior predictive models:

averagespread_prediction <- posterior_predict(
  produce_model, 
  newdata = data.frame(farmprice = c(10, 10), 
            productname = c("Potatoes", "Strawberries")))
mcmc_areas(averagespread_prediction) +   xlab("farmprice") +
  scale_y_discrete(labels = c("Potatoes", "Strawberries"))

Visual examinations: MCMC trace plots, density overlay, autocorrelation function We plot 3 products because there are too many to plot.

par(mfrow = c(2, 2))
mcmc_trace(produce_model, pars=c("productnameStrawberries", "productnamePotatoes", "productnameOranges"), size = 0.5) 

mcmc_dens_overlay(produce_model, pars=c("productnameStrawberries", "productnamePotatoes", "productnameOranges"))

mcmc_acf(produce_model,pars=c("productnameStrawberries", "productnamePotatoes", "productnameOranges"))

Warning: The `facets` argument of `facet_grid()` is deprecated as of ggplot2 2.2.0.
Please use the `rows` argument instead.

The MCMC looks stable!

Numerical examinations: effective sample size and Rhat

neff_ratio(produce_model)

                  (Intercept)                I(1/farmprice)           productnameAvocados 
                      0.10145                       0.68360                       0.12800 
  productnameBroccoli Bunches    productnameBroccoli Crowns         productnameCantaloupe 
                      0.11595                       0.11865                       0.14330 
           productnameCarrots        productnameCauliflower             productnameCelery 
                      0.11755                       0.11685                       0.11770 
      productnameFlame Grapes productnameGreen Leaf Lettuce          productnameHoneydews 
                      0.14440                       0.11955                       0.14430 
   productnameIceberg Lettuce         productnameNectarines            productnameOranges 
                      0.11190                       0.15530                       0.12165 
           productnamePeaches              productnamePlums           productnamePotatoes 
                      0.15785                       0.18305                       0.12455 
  productnameRed Leaf Lettuce    productnameRomaine Lettuce       productnameStrawberries 
                      0.11440                       0.11675                       0.11915 
   productnameThompson Grapes           productnameTomatoes                         sigma 
                      0.17805                       0.14475                       0.81525 

rhat(produce_model)

                  (Intercept)                I(1/farmprice)           productnameAvocados 
                    1.0029506                     1.0000178                     1.0022513 
  productnameBroccoli Bunches    productnameBroccoli Crowns         productnameCantaloupe 
                    1.0025763                     1.0024525                     1.0020801 
           productnameCarrots        productnameCauliflower             productnameCelery 
                    1.0024398                     1.0022483                     1.0024915 
      productnameFlame Grapes productnameGreen Leaf Lettuce          productnameHoneydews 
                    1.0020386                     1.0025774                     1.0022127 
   productnameIceberg Lettuce         productnameNectarines            productnameOranges 
                    1.0026475                     1.0019266                     1.0022810 
           productnamePeaches              productnamePlums           productnamePotatoes 
                    1.0018878                     1.0015216                     1.0024681 
  productnameRed Leaf Lettuce    productnameRomaine Lettuce       productnameStrawberries 
                    1.0024317                     1.0023283                     1.0022952 
   productnameThompson Grapes           productnameTomatoes                         sigma 
                    1.0015973                     1.0026010                     0.9999447 

All neff_ratios are >0.1, and all rhats are 1<rhat<1.05, so we know our MCMC has worked well.

Posterior credible intervals:

posterior_interval(produce_model, prob = 0.90)

                                       5%        95%
(Intercept)                     52.591079   77.33434
I(1/farmprice)                  47.690885   49.65273
productnameAvocados            -61.941457  -32.89030
productnameBroccoli Bunches     58.370767   86.28399
productnameBroccoli Crowns      64.134204   91.84403
productnameCantaloupe           36.806662   67.85688
productnameCarrots             -45.855134  -17.32926
productnameCauliflower          83.521263  111.41925
productnameCelery               71.702124   99.52841
productnameFlame Grapes         -6.789946   23.66940
productnameGreen Leaf Lettuce  161.433185  189.65963
productnameHoneydews            24.871879   56.73218
productnameIceberg Lettuce      59.644561   87.44497
productnameNectarines           68.322882  102.11667
productnameOranges            -179.629902 -148.74820
productnamePeaches              86.123902  119.81075
productnamePlums                26.569034   62.46070
productnamePotatoes            144.654829  172.78067
productnameRed Leaf Lettuce    180.928227  209.23566
productnameRomaine Lettuce     168.767226  196.74644
productnameStrawberries          1.247134   29.09246
productnameThompson Grapes       2.030928   36.92291
productnameTomatoes            102.637038  133.92055
sigma                          120.593366  122.86748

Most of the intervals are not wide and do not include 0, making our model pretty confident.

Posterior predictive check:

pp_check(produce_model)

This doesn’t look very promising. Our model only mildly mimics the structure.

Let’s predict a set of values. Visualizing the posterior predictive model for the averagespread at farmprice = $3 for 5 random products:

products <- sample(unique(produce$productname), 5)
produce_subset <- subset(produce, productname %in% products & farmprice == 3)
new_data <- data.frame(
  productname = products,
  farmprice = c(3, 3, 3, 3, 3)
)
predict_product <- posterior_predict(
  produce_model, 
  newdata = new_data)
head(predict_product)

              1        2        3          4           5
[1,]  -85.87958 316.2543 369.3547  99.758310  120.302357
[2,]   74.26453 294.9136 152.6814 -28.284534  115.777182
[3,]  -52.16765 291.1514 331.7296 161.322007  229.373504
[4,]  230.28737 321.5352 267.5527 241.136481 -166.999399
[5,]  -17.90543 201.8585 386.2392 266.321521 -121.350961
[6,] -197.70137 229.5699 372.7320   5.220148   -9.568815

mcmc_areas(predict_product, prob = 0.8) +
 ggplot2::scale_y_discrete(
   labels = products)

The predictive check demonstrates that our model may not be as accurate as we hope.

Interact Model

We can reason that each scatterplot of farmprice vs averagespread depends on the specific productname; as such, for comparison, we’ll build a model that assumes they interact.

interaction_model <- stan_glm(
  averagespread ~ I(1/farmprice) + productname + productname:I(1/farmprice), 
  data = produce, family = gaussian,
  prior_intercept = normal(25, 5),
  prior = normal(0, 2.5, autoscale = TRUE), 
  prior_aux = exponential(1, autoscale = TRUE),
  chains = 4, iter = 20000*2, seed = 84735)

Posterior summary statistics:

tidy(interaction_model, effects = c("fixed", "aux"))

Visualizing the posterior model structure:

produce %>%
  add_fitted_draws(interaction_model, n = 200) %>%
  ggplot(aes(x = farmprice, y = averagespread, 
             color = productname)) +
    geom_line(alpha = 0.1, aes(y = .value, 
                  group = paste(productname, .draw)))

This model also seems to mimic the original structure.

Plotting posterior predictive models:

averagespread_prediction_2 <- posterior_predict(
  interaction_model, 
  newdata = data.frame(farmprice = c(10, 10), 
            productname = c("Potatoes", "Strawberries")))
mcmc_areas(averagespread_prediction) +   xlab("farmprice") +
  scale_y_discrete(labels = c("Potatoes", "Strawberries"))

Visual examinations: MCMC trace plots, density overlay, autocorrelation function There are too many to list, so we show 3 productnames.

par(mfrow = c(2, 2))
mcmc_trace(interaction_model, pars=c("productnameStrawberries", "productnamePotatoes", "productnameOranges"), size = 0.5) 

mcmc_dens_overlay(interaction_model, pars=c("productnameStrawberries", "productnamePotatoes", "productnameOranges"))

mcmc_acf(interaction_model,pars=c("productnameStrawberries", "productnamePotatoes", "productnameOranges"))

The MCMC’s are looking stable!

Numerical examinations: effective sample size and Rhat

neff_ratio(interaction_model)

                                 (Intercept)                               I(1/farmprice) 
                                   0.0654875                                    0.0641500 
                         productnameAvocados                  productnameBroccoli Bunches 
                                   0.1051000                                    0.0761125 
                  productnameBroccoli Crowns                        productnameCantaloupe 
                                   0.0775625                                    0.0769750 
                          productnameCarrots                       productnameCauliflower 
                                   0.0863375                                    0.0716250 
                           productnameCelery                      productnameFlame Grapes 
                                   0.0768750                                    0.1148875 
               productnameGreen Leaf Lettuce                         productnameHoneydews 
                                   0.0780500                                    0.0769000 
                  productnameIceberg Lettuce                        productnameNectarines 
                                   0.0747500                                    0.1050625 
                          productnameOranges                           productnamePeaches 
                                   0.0714000                                    0.0935250 
                            productnamePlums                          productnamePotatoes 
                                   0.1191250                                    0.0793875 
                 productnameRed Leaf Lettuce                   productnameRomaine Lettuce 
                                   0.0796500                                    0.0738000 
                     productnameStrawberries                   productnameThompson Grapes 
                                   0.0766875                                    0.1187750 
                         productnameTomatoes           I(1/farmprice):productnameAvocados 
                                   0.0937250                                    0.0765875 
  I(1/farmprice):productnameBroccoli Bunches    I(1/farmprice):productnameBroccoli Crowns 
                                   0.0649125                                    0.0655250 
        I(1/farmprice):productnameCantaloupe            I(1/farmprice):productnameCarrots 
                                   0.0645750                                    0.0648875 
       I(1/farmprice):productnameCauliflower             I(1/farmprice):productnameCelery 
                                   0.0646375                                    0.0651375 
      I(1/farmprice):productnameFlame Grapes I(1/farmprice):productnameGreen Leaf Lettuce 
                                   0.0875625                                    0.0647250 
         I(1/farmprice):productnameHoneydews    I(1/farmprice):productnameIceberg Lettuce 
                                   0.0644625                                    0.0647375 
        I(1/farmprice):productnameNectarines            I(1/farmprice):productnameOranges 
                                   0.0681125                                    0.0641750 
           I(1/farmprice):productnamePeaches              I(1/farmprice):productnamePlums 
                                   0.0662875                                    0.0728000 
          I(1/farmprice):productnamePotatoes   I(1/farmprice):productnameRed Leaf Lettuce 
                                   0.0714625                                    0.0647625 
   I(1/farmprice):productnameRomaine Lettuce       I(1/farmprice):productnameStrawberries 
                                   0.0643375                                    0.0738750 
   I(1/farmprice):productnameThompson Grapes           I(1/farmprice):productnameTomatoes 
                                   0.0855625                                    0.0670500 
                                       sigma 
                                   0.6506750 

rhat(interaction_model)

                                 (Intercept)                               I(1/farmprice) 
                                   1.0008736                                    1.0009001 
                         productnameAvocados                  productnameBroccoli Bunches 
                                   1.0004240                                    1.0007940 
                  productnameBroccoli Crowns                        productnameCantaloupe 
                                   1.0006821                                    1.0008274 
                          productnameCarrots                       productnameCauliflower 
                                   1.0006600                                    1.0008309 
                           productnameCelery                      productnameFlame Grapes 
                                   1.0006684                                    1.0005168 
               productnameGreen Leaf Lettuce                         productnameHoneydews 
                                   1.0007473                                    1.0007326 
                  productnameIceberg Lettuce                        productnameNectarines 
                                   1.0008062                                    1.0004947 
                          productnameOranges                           productnamePeaches 
                                   1.0007899                                    1.0007152 
                            productnamePlums                          productnamePotatoes 
                                   1.0003909                                    1.0008420 
                 productnameRed Leaf Lettuce                   productnameRomaine Lettuce 
                                   1.0006249                                    1.0007640 
                     productnameStrawberries                   productnameThompson Grapes 
                                   1.0007227                                    1.0005306 
                         productnameTomatoes           I(1/farmprice):productnameAvocados 
                                   1.0006794                                    1.0006770 
  I(1/farmprice):productnameBroccoli Bunches    I(1/farmprice):productnameBroccoli Crowns 
                                   1.0009105                                    1.0008701 
        I(1/farmprice):productnameCantaloupe            I(1/farmprice):productnameCarrots 
                                   1.0009098                                    1.0008901 
       I(1/farmprice):productnameCauliflower             I(1/farmprice):productnameCelery 
                                   1.0009173                                    1.0008738 
      I(1/farmprice):productnameFlame Grapes I(1/farmprice):productnameGreen Leaf Lettuce 
                                   1.0006768                                    1.0008951 
         I(1/farmprice):productnameHoneydews    I(1/farmprice):productnameIceberg Lettuce 
                                   1.0008971                                    1.0009061 
        I(1/farmprice):productnameNectarines            I(1/farmprice):productnameOranges 
                                   1.0008294                                    1.0008960 
           I(1/farmprice):productnamePeaches              I(1/farmprice):productnamePlums 
                                   1.0009041                                    1.0007183 
          I(1/farmprice):productnamePotatoes   I(1/farmprice):productnameRed Leaf Lettuce 
                                   1.0008966                                    1.0008714 
   I(1/farmprice):productnameRomaine Lettuce       I(1/farmprice):productnameStrawberries 
                                   1.0008922                                    1.0007620 
   I(1/farmprice):productnameThompson Grapes           I(1/farmprice):productnameTomatoes 
                                   1.0007393                                    1.0009009 
                                       sigma 
                                   0.9999916 

While all rhats are 1<rhat<1.05, most of the neff_ratios are less than 0.1, which raises suspicions. Nevertheless, we’ll continue.

Posterior credible intervals:

posterior_interval(interaction_model, prob = 0.90)

                                                       5%          95%
(Intercept)                                   -48.0849972    5.9595760
I(1/farmprice)                                131.5788743  196.1286292
productnameAvocados                           -64.9287114    6.1223660
productnameBroccoli Bunches                    -8.2661768   50.6393634
productnameBroccoli Crowns                    -67.2684961   -7.7113733
productnameCantaloupe                         140.7116421  199.5125428
productnameCarrots                            170.3538339  233.2132419
productnameCauliflower                         91.9004828  148.5577839
productnameCelery                             -64.1529247   -5.6126519
productnameFlame Grapes                        -0.3694624   74.6099091
productnameGreen Leaf Lettuce                 -17.7256860   41.3959064
productnameHoneydews                          147.6527766  206.3706225
productnameIceberg Lettuce                    -51.3726753    6.6361871
productnameNectarines                         -80.1630429  -10.1574252
productnameOranges                            204.3599478  261.0656192
productnamePeaches                            -69.8626095   -3.1717708
productnamePlums                               -9.5461900   65.7297610
productnamePotatoes                           -15.9331257   44.5522883
productnameRed Leaf Lettuce                    11.1404345   71.2498646
productnameRomaine Lettuce                     23.9551728   81.9085423
productnameStrawberries                       -53.6868233    5.2961016
productnameThompson Grapes                    -41.6769656   36.1341603
productnameTomatoes                           -35.7999253   29.8224292
I(1/farmprice):productnameAvocados           -100.0683852  -28.0259388
I(1/farmprice):productnameBroccoli Bunches    -90.2873195  -25.1477241
I(1/farmprice):productnameBroccoli Crowns     -52.3189675   13.0967386
I(1/farmprice):productnameCantaloupe         -155.4599838  -90.6705800
I(1/farmprice):productnameCarrots            -181.8281270 -116.7660726
I(1/farmprice):productnameCauliflower        -119.4527145  -54.6587844
I(1/farmprice):productnameCelery              -65.6026194   -0.4458604
I(1/farmprice):productnameFlame Grapes       -100.9623967  -23.2382809
I(1/farmprice):productnameGreen Leaf Lettuce  -75.3185316  -10.4345943
I(1/farmprice):productnameHoneydews          -160.8630953  -95.9622282
I(1/farmprice):productnameIceberg Lettuce     -83.0342194  -18.0872839
I(1/farmprice):productnameNectarines          -53.5348779   13.5963644
I(1/farmprice):productnameOranges            -189.3568623 -124.7797404
I(1/farmprice):productnamePeaches             -61.2353438    4.8749041
I(1/farmprice):productnamePlums               -93.9098907  -24.4447085
I(1/farmprice):productnamePotatoes             71.6730073  140.7311885
I(1/farmprice):productnameRed Leaf Lettuce    -80.2638983  -15.3207883
I(1/farmprice):productnameRomaine Lettuce     -86.5549448  -21.7666723
I(1/farmprice):productnameStrawberries          9.8136059   79.8301458
I(1/farmprice):productnameThompson Grapes     -64.4795998   12.7960180
I(1/farmprice):productnameTomatoes            -52.7397275   13.7217412
sigma                                          85.3865777   87.0003819

Most of the credible intervals do not include 0, which is a good sign. However, quite a few of them are very wide, which raises some alarms.

Posterior predictive check:

pp_check(interaction_model)

This model seems to fit better than our first model did, although it’s not as stable as the first model was. Visualizing the posterior predictive model for the averagespread at farmprice = $3 for 5 random products:

products <- sample(unique(produce$productname), 5)
produce_subset <- subset(produce, productname %in% products & farmprice == 3)
new_data <- data.frame(
  productname = products,
  farmprice = c(3, 3, 3, 3, 3)
)
predict_product <- posterior_predict(
  interaction_model, 
  newdata = new_data)
head(predict_product)

             1         2         3         4         5
[1,]  27.85177  14.57094 254.50606  32.15601 203.43253
[2,] 137.30504 -50.27387  88.36031 -42.65681 175.63443
[3,]  47.82571 129.56534 246.68158 218.31116 206.88012
[4,] 192.59296 -20.23407 123.40767 157.71656 125.15852
[5,] 139.56300 -21.97496 329.65007 137.54000  76.73732
[6,] 142.44043 106.42295 171.00168  90.55963  70.26058

mcmc_areas(predict_product, prob = 0.8) +
 ggplot2::scale_y_discrete(
   labels = products)

Let’s perform a loo comparison.

Predictive Accuracy, Comparing Models

Predictive accuracy: ELPD Calculate the cross-validated expected log-predictive densities (ELPD) and compare with the interaction_model:

elpd_linear <- loo(produce_model)
elpd_interaction <- loo(interaction_model)
loo_compare(elpd_linear, elpd_interaction)

                  elpd_diff se_diff
interaction_model     0.0       0.0
produce_model     -5378.2     128.3

The interaction model performs significantly better against the data than the linear model does. This can be explained by the fact that the trend of the scatterplot changes depending on the productname. Each product has a unique relationship between farmprice and averagespread, so it makes sense that the interaction model would perform better.

Citations

Dataset downloaded from Kaggle at https://www.kaggle.com/datasets/everydaycodings/produce-prices-dataset by KUMAR SAKSHAM; originally sourced from http://www.producepriceindex.com.